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Mirrors > Home > MPE Home > Th. List > annotanannot | Structured version Visualization version GIF version |
Description: A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.) |
Ref | Expression |
---|---|
annotanannot | ⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar 530 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
2 | 1 | bicomd 222 | . . 3 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓)) |
3 | 2 | notbid 318 | . 2 ⊢ (𝜑 → (¬ (𝜑 ∧ 𝜓) ↔ ¬ 𝜓)) |
4 | 3 | pm5.32i 576 | 1 ⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 398 |
This theorem is referenced by: suppcoss 8142 clwwlknclwwlkdif 28972 0nn0m1nnn0 33767 |
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